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You might want to double check the question, because there is no real pair of numbers that holds those properties. There are, however, complex numbers that do:

$y = 10 - x$

$x^3 + (10-x)^3 = 162 +x^2 + (10-x)^2$

$x^3 + (10-x)^2(10-x) = 162 + x^2 + (100 - 20x + x^2)$

$x^3 + (100-20x+x^2)(10-x) = 2x^2-20x+262$

$x^3 + (1000 -200x+10x^2-100x+20x^2-x^3) = 2x^2-20x+262$

$30x^2-300x+1000 = 2x^2 - 20x + 262$

$15x^2-150x+500=x^2-10x+131$

$14x^2-140x+369 = 0$

$x^2-10x+\frac{369}{14} = 0$

$x^2 - 10x +\frac{350}{14} = -\frac{19}{14}$

$x^2 - 10x + 25 = -\frac{19}{14}$

$(x-5)^2 = -\frac{19}{14}$

$x-5 = \sqrt{\frac{19}{14}}i$

$x = 5 + \sqrt{\frac{19}{14}}i$

$y = 10 - (5 + \sqrt{\frac{19}{14}}i)$

$y = 5 - \sqrt{\frac{19}{14}}i$

Therefore, one ordered pair of complex numbers that works is 5 + sqrt(19/14)i and 5 - sqrt(19/14)i

Please let me know if I had made any errors!